Let X : = {(an)n>1| an ER}. Define d : X x X - [0, too) by d((an) n21, (bn)n21) := min {m E N : am * bm} if (an)n21 * (bn)n>1, and otherwise d( (an)n21, (bn)n21) = 0. i.) (2 points) Prove that d defines a metric on X. ii.) (3 points) Let A C X be the set of all sequences (an)n>1 which EI- THER begin with 0, 1, 2 (namely: a1 = 0 and a2 = 1 and a3 = 2) OR begin with 3, 4, 5, 6 (namely: a1 = 3 and a2 = 4 and a3 = 5 and a4 = 6). Prove that A is open in (X, d). iii.) (4 points) Prove that the set of all constant sequences is closed in (X, d)